Hypothesis Testing

An obvious question is how to carry out various diagnostic tests done in the parametric econometrics within the nonparametric and semiparametric models. Several papers have appeared in the recent literature which deal with this issue. We present them here and show their links.

Подпись: T1 = Подпись: (PRSS - NPRSS) NPRSS Подпись: PRSS 1 X U2 — X Й2 NPRSS — _ XB '

First consider the problem of testing a specified parametric model against a nonparametric alternative, H0 : /(p, x) = E(yi | xi) against H1 : m(x) = E(yi | xi). The idea behind the Ullah (1985) test statistic is to compare the parametric RSS (PRSS) XЙ2, x = Vi – /(S, xi) with the nonparametric RSS (NPRSS), XB2, where щ = – m(xi). His test statistic is

or simply T * = (PRSS – NPRSS), and reject the null hypothesis when T1 is large. л/П T1 has a degenerage distribution under H0. Lee (1994) uses density weighted
residuals and compares Xwfi} with X A2 to avoid degeneracy, for other procedures see Pagan and Ullah (1999). Pagan and Ullah (1999) also indicate the normalizing factor needed for the asymptotic normality of T1. An alternative suggested here is the following nonparametric bootstrap method:

1. Generate the bootstrap residuals u* from the centered residuals (fl; – u) where й is the average of й.

2. Generate y* = f (S, x,) + u* from the null model.

3. Using the bootstrap sample x;, y*, i = 1,…, n, estimate m(x) nonparametrically, say and m*(x,), and obtain the bootstrap residual A* = y* – m*(x,).

4. Calculate the bootstrap test statistic T* = (Xй2 – Xй*)2/Хй*2.

5. Repeat steps (1) to (4) B times and use the empirical distribution of T* as the null distribution of T*.

An alternative is to use wild bootstrap method or pivotal bootstrap which will preserve the conditional heteroskedasticity in the original residuals. Another alternative is to use the block bootstrap method (Buhlman and Kunsch, 1995).

An alternative test statistic is based on comparing the parametric fit with the nonparametric fit. Defining a(x) as a smooth weight function, this test statistic is

Подпись:(f (S, x,) – m(x,))2a(x,)dF(x),

where F is the empirical distribution function, see Ullah (1985) and Gozalo

(1995) , and Ai’t-Sahalia et al. (1998) who also indicate that f(S, x,) and m(x,) can also be replaced by f(S, x,) – ff (S, x,)f(x,)dxi and m(xi) by m(xi) – fm(x,)f(x)dxi without affecting the results in practice and provide the asymptotic normality of

nhq/2T2.

Подпись: T3 = Подпись: (f(S, x) - m(x))2a(x)dx x Подпись: [Е(й | x)]2a(x)dx, x

Hardle and Mammen (1993) suggest a weighted integrated square difference between the nonparametric estimator and the kernel smoothed parametric estimator f ф, x,) = E(f(S, xf)| x,-) which can be calculated by the LLS procedures with y, replaced by f (S, x,). This is

where E(m|x) = E( y|x) – E(f(S, x)|x) = m(x) – f(S, x). It has been shown in Rahman and Ullah (1999) that the use of the kernel smoothed estimator f(S, x) gives better size and power performances of the tests compared to the case of using f(S, x). T3 is similar to T2 if we write a(x) = a(x)f-1(x)dF(x) and use the empirical distribution. This test statistic is computationally involved. In view of this, Li and Wang (1998) and Zheng (1996) proposed a conditional moment test (CMT) which is easy to calculate, and has a better power performance. Its form is

image493

1 n

X Ui(m(x,) – x!))f(x!).

n i

This statistic is based on the idea that under the null E(uixl) = E[uiE(uix)] = E[(Em;|x;)2] = E[uiE(uix)a(x)] = 0 for any positive a(x). T4 is an estimate of E[u;E(u; x;)a(x;)] for a(x,) = f(x). Eubank and Spiegelman (1990), however, tests for E[(Eu; x;)2] = 0 using a series type estimator of E(u; x;). The test statistic T4 has nhq/2 rate of convergence to normality.

All the above nonparametric tests are generally calculated with the leave-one – out estimators of m(xi) = m-i(x) and the weight a(x.) = f(xi) = f-i(xI). Theoretically, the use of leave-one-out estimators helps to get asymptotic normality centered at zero. The tests are consistent model specification tests in the sense that their power goes to one as n ^ ™ against all the alternatives. The usual parametric specification tests are however consistent against a specified alternative. An approach to developing consistent model specification test, without using any nonparametric estimator of m(x), is the CMT due to Bierens-Newey-Tauchen, see Pagan and Ullah (1999) for details. An important difference between Bierens – type tests and the tests T1 to T6 is the treatment of h. While T1 to T6 tests consider h ^ 0 as n ^ ж, Bierens (1982) type tests treat h to be fixed which makes the asymptotic distribution of their tests to be nonnormal but can detect the Pitman’s
local alternative that approach the null at the rate O(n~1/2) compared to the slower rate of O((nhq/2)-1/2) of T1 to T6. However, Fan and Li (1996) indicate that under high frequency type local alternatives the tests with vanishing h may be more powerful than tests based on fixed h. For asymptotic normality of the tests T1 to T6 for the dependent observations, see Li (1997).

The test statistics T1 to T6 can also be used for the problem of variable selec­tions. For example, testing H0 : m(x,) = m(xi1, xi2) = m(xi1) against H1 : m(x) Ф m(xa) can be carried out by calculating the difference between the NPRSS due to m(xi1, x12) and the NPRSS due to m(xi1), or using T2 = n-1’L(m(xi1, xi2) – m(xi1))2a(x,) test, see Ai’t-Sahalia et al. (1998) for asymptotic normality. An alternative is suggested in Racine (1997). We can also do the diagnostics for variable selection by using the goodness of fit measures described in Section 2.1; in addition see Vien (1994) where the cross-validation method has been used.

The tests T1 to T6 can also be extended to do nonnested testing (Delgado and Mora, 1998), testing for parametric and semiparametric models yi = f ф, x;) + Xm(xJ) + ui, (Li, 1997; Fan and Li, 1996) and single index models, Ai’t-Sahalia et al., 1998). Finally, for testing the restrictions on the parameters, testing hetero – skedasticity, and testing serial correlation in the parametric model yi = f(P, x;) + ui with the unknown form of density, see Gonzalez-Rivera and Ullah (1999) where they develop the semiparametric Rao-score test (Lagrange multiplier) with the unknown density replaced by its kernel estimator. Also see Li and Hsiao (1998) for a semiparametric test of serial correlation.

Note

* The author is thankful to two referees for their constructive comments and suggestions. The research support from the Academic Senate, UCR, is gratefully acknowledged.

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